Invariant subsets of expanding mappings of the circle

Abstract

The continuity of Hausdorff dimension of closed invariant subsets K of a C2-expanding mapping g of the circle is investigated. If g/K satisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that if f is a piecewise monotone continuous mapping of a compact interval and φ a continuous function with P(f,φ)> sup(φ), then the pressure P(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.